Transmit and receive beamforming is an attractive, low-complexity technique for exploiting the significant diversity that is available in multiple-input and multiple-output (MIMO) wireless systems. Unfortunately, optimal performance requires either complete channel knowledge or knowledge of the optimal beamforming vector which is difficult to realize in practice. In this paper, we propose a quantized maximum signal-to-noise ratio beamforming technique where the receiver only sends the label of the best weight vector in a predetermined codebook to the transmitter. We develop optimal codebooks for i.i.d. Rayleigh fading matrix channels. We derive the distribution of the optimal transmit weight vector and use this result to bound the SNR degradation as a function of the quantization. A codebook design criterion is proposed by exploiting the connections between the quantization problem and Grassmannian line packing. The design criterion is flexible enough to allow for side constraints on the codebook vectors. Bounds on the maximum distortion with the resulting Grassmannian codebooks follow naturally from the nature of the code. A proof is given that any system using an overcomplete codebook for transmit diversity and maximum ratio combining obtains a diversity on the order of the product of the number of transmit and receive antennas. Bounds on the loss in capacity due to quantization are derived. Monte Carlo simulations are presented that compare the symbol error probability for different quantization strategies.
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